Title:

Kind
Code:

A1

Abstract:

The diffusion coefficients and relaxation times of mixtures of alkanes follow simple scaling laws based on the chain length of the constituents and the mean chain length of the mixture. These scaling laws are used to determine chain sizes in a mixture from the distribution of the diffusion coefficients. These scaling laws can be used to determine the mean chain lengths (or chain lengths) of a sample (alkanes or mixtures of alkanes) and therefore the constituents of the sample.

Inventors:

Freed, Denise (Mount Kisco, NY, US)

Application Number:

10/864124

Publication Date:

12/16/2004

Filing Date:

06/09/2004

Export Citation:

Assignee:

SCHLUMBERGER TECHNOLOGY CORPORATION, Incorporated in the State of Texas (Ridgefield, CT)

Primary Class:

International Classes:

View Patent Images:

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Primary Examiner:

GAKH, YELENA G

Attorney, Agent or Firm:

SCHLUMBERGER-DOLL RESEARCH (Houston, TX, US)

Claims:

1. A method for determining the characteristics of a fluid sample, comprising: a. obtaining measurements on a plurality of calibration samples having one or more known constituents, wherein the calibration samples have differing mean chain lengths, and wherein said measurements are selected from the group consisting of diffusion measurements and relaxation measurements; and b. determining the scaling law of said plurality of fluid samples using as a function of chain length said measurements of (a).

2. The method of claim 1, further comprising: c. obtaining measurements on a sample under investigation, wherein said measurements are selected from the group consisting of diffusion measurements and relaxation measurements; and d. identifying the constituents of said sample under investigation by applying the scaling law of (b) to the measurements of (c).

3. The method of claim 1, wherein the measurements of (a) are obtained using nuclear magnetic resonance techniques.

4. The method of claim 2, wherein the measurements of (c) are obtained using nuclear magnetic resonance techniques.

5. The method of claim 1, wherein the measurements of (a) are performed at a first temperature and a reference pressure.

6. The method of claim 2, wherein the measurements of (c) are performed at said first temperature and said reference pressure.

7. The method of claim 5, wherein determining the scaling law includes performing a two-parameter fit of the function.

8. The method of claim 7, wherein the two parameters of said two-parameter fit include the slope and intercept.

9. The method of claim 6, wherein said first temperature and said reference pressure are approximately equal to the temperature and pressure of the sample under investigation at the time of sampling.

10. The method of claim 2, wherein (d) further comprises determining the mean chain length of the constituents of said sample under investigation.

11. The method of claim 2, wherein (d) further comprises determining the chain length distribution of said sample under investigation.

12. The method of claim 2, further comprising: e. obtaining the distribution of chain lengths of the sample under investigation using diffusion measurements; f. obtaining the distribution of chain lengths of the sample under investigation using relaxation measurements; g. comparing the distribution of chain lengths of (e) to the distribution of chain lengths of (f) to determine the presence of asphaltenes in said sample under investigation.

13. The method of claim 2, further comprising repeating (c) and (d) for one or more additional samples under investigation.

14. The method of claim 13, further comprising: determining the mean chain length of each additional sample under investigation.

15. The method of claim 13, further comprising: determining the distribution of chain lengths of each additional sample under investigation.

16. The method of claim 14, further comprising comparing the mean chain length of each sample under investigation.

17. The method of claim 15, further comprising: comparing the distribution of chain lengths of each additional sample under investigation.

18. The method of claim 15, wherein each sample under investigation is fluid from one or more regions of earth formation.

19. The method of claim 17, further comprising comparing the distribution of chain lengths of each sample under investigation to determine the composition gradient between said samples under investigation.

20. The method of claim 19, wherein said samples under investigation are at locations along a common borehole.

21. The method of claim 19, wherein said samples under investigation are at two or more different boreholes.

22. A method for determining the characteristics of a fluid sample, comprising: a. obtaining measurements on a plurality of calibration samples having one or more known constituents at more than one temperature and a reference pressure, wherein the calibration samples have differing mean chain lengths, and wherein said measurements are selected from the group consisting of diffusion measurements and relaxation measurements; and b. determining the scaling law of said plurality of calibration samples as a function of chain length and temperature, using the measurements of (a).

23. The method of claim 22, further comprising: c. performing measurements on a sample under investigation at said reference pressure and at a temperature within or near the range of temperatures of (a), wherein said measurements are selected from the group consisting of relaxation measurements and diffusion measurements; d. identifying the constituents of said sample under investigation by applying the scaling law of (b) to the measurements of (c).

24. The method of claim 22, wherein the scaling law of (b) is a function of chain length and mean chain length.

25. The method of claim 22, wherein the measurements of (a) are obtained using nuclear magnetic resonance techniques.

26. The method of claim 23, wherein the measurements of (c) are obtained using nuclear magnetic resonance techniques.

27. The method of claim 26, wherein (d) further comprises determining the distribution of chain lengths of the constituents of said sample under investigation.

28. The method of claim 23, further comprising: e. obtaining the distribution of chain lengths of the sample under investigation using diffusion measurements; f. obtaining the distribution of chain lengths of the sample under investigation using relaxation measurements; g. comparing the distribution of chain lengths of (e) to the distribution of chain lengths of (f) to determine the presence of asphaltenes in said sample under investigation.

29. The method of claim 23, further comprising repeating (c) and (d) for one or more additional samples under investigation.

30. The method of claim 29, further comprising: determining the distribution of chain lengths of each additional sample under investigation.

31. The method of claim 30, further comprising comparing the distribution of chain lengths of each sample under investigation.

32. The method of claim 31, wherein each sample under investigation is fluid from one or more regions of earth formation.

33. The method of claim 32, further comprising comparing the chain lengths of each sample under investigation to determine the composition gradient between said samples under investigation.

34. The method of claim 33, wherein said samples under investigation are at locations along a common borehole.

35. The method of claim 33, wherein said samples under investigation are at two or more different boreholes.

36. The method of claim 22, wherein (b) further includes performing a four parameter fit in terms of mean chain length and temperature.

37. A method for determining the characteristics of a fluid sample, comprising: a. obtaining measurements of a plurality of calibration samples at a first temperature and a reference pressure, wherein the calibration samples have differing mean chain lengths, and wherein said measurements are selected from the group consisting of diffusion measurements and relaxation measurements; b. determining the density of more than one pure alkane or mixtures of alkanes at the first temperature and the reference pressure, wherein density is determined as a function of mean chain length; c. obtaining measurements of a sample under investigation at the first temperature and a second pressure, wherein said measurements are selected from the group consisting of diffusion measurements and relaxation measurements; d. determining the density of the sample under investigation at the first temperature and the second pressure; e. applying the density function of (c) to the density measurements of (d) and using the measurements of (a) to determine the scaling law at the second pressure in terms of chain length; f. applying the scaling law of (e) to the data of (c) to determine the composition of the sample under investigation.

38. The method of claim 37, wherein said density measurements of (b) are obtained from standard look-up tables.

39. The method of claim 37, wherein (f) further comprises determining the distribution of chain lengths of the constituents of said sample under investigation.

40. The method of claim 37, wherein the density measurement of (d) is selected from the group consisting of mass density, carbon density, and hydrogen index.

41. A method for determining the characteristics of a fluid sample, comprising: a. obtaining measurements of a plurality of calibration samples at a reference pressure and at more than one temperature, wherein the calibration samples have differing mean chain lengths, and wherein said measurements are selected from the group consisting of diffusion measurements and relaxation measurements; b. determining the density of more than one pure alkane or mixtures of alkanes at the reference pressure and at a temperature within or near the range of temperatures in (a), wherein density is determined as a function of mean chain length; c. obtaining measurements of a sample under investigation at a second pressure and at a temperature within or near the range of temperatures in (a), wherein said measurements are selected from the group consisting of diffusion measurements and relaxation measurements; d. determining the density of the sample under investigation at the second pressure and at a temperature within or near the range of temperatures in (a); e. applying the density function of (c) to the density measurements of (d) and using the measurements of (a) to determine the scaling law at the second pressure in terms of chain length; f. applying the scaling law of (e) to the data of (c) to determine the composition of the sample under investigation.

42. The method of claim 41, wherein said density measurements of (b) are obtained from standard look-up tables.

43. The method of claim 41, wherein (f) further comprises determining the distribution of chain lengths of the constituents of said sample under investigation.

44. The method of claim 41, wherein the density measurement of (d) is selected from the group consisting of mass density, carbon density, and hydrogen index.

45. A method for determining the characteristics of a fluid sample, comprising: a. obtaining measurements of a plurality of calibration samples at a first temperature and a reference pressure, wherein the calibration samples have differing mean chain lengths, and wherein said measurements are selected from the group consisting of diffusion measurements and relaxation measurements; b. obtaining measurements of said sample under investigation at the first temperature and a second pressure, wherein said measurements are selected from the group consisting of diffusion measurements and relaxation measurements; c. determining the relationship of volume of one or more alkanes or mixtures of alkanes to (i) the mean chain length at the first temperature and the reference pressure and (ii) the mean chain length at the first temperature and a second pressure; d. determining the scaling law as a function of chain length, using the functions of (c) and the measurements of (a); e. applying the scaling law of (d) to the measurements of (b) to determine the composition of the sample under investigation.

46. The method of claim 45, wherein the scaling law of (d) is a function of chain length and mean chain length.

47. The method of claim 45, wherein said volume of (c) are obtained using standard look-up tables.

48. The method of claim 45, wherein the volumes of (c) are selected from the group consisting of volume per mole, volume per hydrogen atom, and volume per carbon atom.

49. The method of claim 45, wherein (e) further comprises determining the distribution of chain lengths of the constituents of said sample under investigation.

50. A method for determining the characteristics of a fluid sample, comprising: a. obtaining measurements of a plurality of calibration samples at more than one temperature and a reference pressure, wherein the calibration samples have differing mean chain lengths, and wherein said measurements are selected from the group consisting of diffusion measurements and relaxation measurements; b. obtaining measurements of said sample under investigation at a temperature within or near the range of temperatures of the measurements of (a) and at a second pressure, wherein said measurements are selected from the group consisting of diffusion measurements and relaxation measurements; c. determining the relationship of volume of one or more alkanes or mixtures of alkanes to (i) the mean chain length at the reference pressure and (ii) the mean chain length at the second pressure; d. determining the scaling law in terms chain length at the second pressure using the functions of (c) and the measurements of (a); e. applying the scaling law of (d) with the measurements of (b) to determine the composition of the sample under investigation.

51. The method of claim 50, wherein the scaling law of (d) is a function of chain length and mean chain length.

52. The method of claim 50, wherein said volume functions of (c) are obtained using standard look-up tables.

53. The method of claim 50, wherein the volumes of (c) are selected from the group consisting of volume per mole, volume per hydrogen atom, and volume per carbon atom.

54. The method of claim 50, wherein (e) further comprises determining the distribution of chain lengths of the constituents of said sample under investigation.

Description:

[0001] The present invention claims priority to co-pending U.S. Provisional Patent Application No. 60/477,515 filed Jun. 11, 2003 and co-pending U.S. Provisional Patent Application No. 60/496,799 filed Aug. 21, 2003. Both of these provisional patent applications are incorporated by reference herein in their entireties.

[0002] The present invention relates to a method of modeling alkanes and, more particularly, to a method of determining the constituents of an oil mixture and its viscosity.

[0003] It is well known that the self-diffusion coefficient of a molecule is related in some way to its size. For a hard sphere in a fluid with viscosity η_{s}

[0004] where r is the radius of the sphere and η_{s }_{i }^{th }

[0005] where r_{i }^{th }_{i}_{i }_{s}

[0006] An application of the hard sphere model to oils may be found in Freedman et al. “A New NMR Method of Fluid Characterization in Reservoir Rocks: Experimental Confirmation and Simulation Results,” paper SPE 63214 presented at the 2000 SPE Annual Technical Conference and Exhibition, Dallas, 1-4 October; Lo et al. “Relaxation Time and Diffusion Measurements of Methane and Decane Mixtures,” The Log Analyst (November-December 1998); and Lo et al. “Correlations of NMR Relaxation Time with Viscosity, Diffusivity, and Gas/Oil Ratio of Methane/Hydrocarbon Mixtures,” in

[0007] As shown in

[0008] However, the hard sphere model is not adequate for describing more complicated molecules, such as oils, which are floppy chains. One example of the failure of the hard sphere model is evidenced by the measurements of diffusion and viscosity in alkanes and oils. In plots of log D versus log kT/η (see

[0009] As will be shown below, elevated temperature and pressure may influence the modeling of complicated oils. Applications of the hard sphere model to oils do not adequately account for the effects of temperature and pressure on the diffusion coefficients and relaxation times.

[0010] Accordingly, it is one object of the present invention to provide a method to more appropriately model oils and oil mixtures.

[0011] It is another object of the present invention to provide a model that accounts for the temperature and pressure dependence of the diffusion coefficient and relaxation times over a wide range of temperatures and pressures.

[0012] Accordingly, the present inventor has discovered that diffusion coefficients and relaxation times of mixtures of alkanes follow simple scaling laws based on the chain length of the constituents and the mean chain length of the mixture. These scaling laws are used to determine chain sizes in a mixture from the distribution of the diffusion coefficients. These scaling laws can be used to determine the mean chain lengths (or chain lengths) in live oils as well as the viscosity of a mixture.

[0013] In addition, the present invention characterizes the relationship between diffusion coefficients D_{i }_{1i }_{2i }_{i}_{1i}_{2i }_{i}_{1i }_{2i }_{0}_{i}_{1i}_{2i }_{i}_{1i}_{2i }

[0014] Accordingly, expressions for diffusion coefficient and relaxation times may be determined as a function of chain lengths at elevated pressures and temperatures using: (1) density data for pure alkanes at the desired pressures and temperatures and (2) data on diffusion coefficients or relaxation times for pure or mixed alkanes at one reference pressure and several temperatures. Preferably, this data spans the range of chain lengths or densities of interest. Once the relation between D, T_{1}_{2 }

[0015] This method is applicable for a wide range of temperatures and pressures and for chain lengths less than the entanglement length. For pressures above about 100 MPa careful selection of the reference pressure is recommended. It is also noted that for pressures above about 100 MPa the slope of the calibration curve may begin to change and should be accounted for. If asphaltenes or a large amount of aromatics are expected to be present, it may be preferable to obtain one or more additional measurements to determine the type of molecules in the mixtures. A difference between the chain length distribution found by measuring the distributions of diffusion coefficients and relaxation times may identify the presence of asphaltenes. In addition, the difference between T_{1 }_{2 }

[0016] The method of the present invention can be quite useful in detecting gradients in composition along a well or between wells. If the oil is of the type that varies with temperature and pressure as in the alkane model, then the NMR derived distribution could be calibrated with laboratory oil measurements at a few places, and points between the measured NMR distributions would indicate composition gradients. In the absence of lab measurements, the NMR distributions can show composition gradients; however, lab measurements would verify that the temperature and pressure dependence is of the expected form.

[0017] One advantage of obtaining the composition from the diffusion or relaxation measurements is that it is complementary to optical measurements, first, because they measure different types of physics, and second, because they NMR measurement can give some detailed information about the molecules with longer chain lengths, while optical measurements can give details about the exact methane content and the presence of other gases. Methane can affect the scaling law for the NMR relaxation times, while the presence of other gases such as CO_{2 }_{1 }

[0018] Because the NMR measurements are sensitive to larger particles, it can also be useful for detecting phase changes. As waxes or asphaltenes start to aggregate or precipitate, they should appear in the chain length distribution as much larger particles, which can signal a phase change as the temperature or pressure is changed.

[0019] Accordingly, the present invention provides a method of determining the constituents of an oil mixture and its viscosity. The method of the present invention includes using commonly known nuclear magnetic resonance techniques to determine the diffusion distribution of a mixture and, using polymer models, correlating this diffusion distribution to chain length of the constituents, the mean chain length of the mixture, or its viscosity.

[0020] Accordingly, a first embodiment of the present invention is a method for determining the characteristics of a fluid sample, comprising: (a) obtaining measurements (diffusion or relaxation measurements) on a plurality of calibration samples having one or more known constituents; and (b) determining the scaling law of said plurality of fluid samples using as a function of chain length said measurements of (a). To create an accurate calibration, the calibration samples should have a variety of mean chain lengths. In addition, the calibration samples may be pure alkanes or mixtures of alkanes, or a combination thereof. Once this calibration is determined, the constituents of a sample under investigation may be determined by obtaining either diffusion measurements and relaxation measurements, depending on the measurements made in (a) above and then applying the scaling law of (b) to these measurements. It is noted that the calibration does not need to be redone for each sample; once a calibration is performed, it may be reused for other samples. In one application of this embodiment, the calibration measurements and the sample measurements are performed at a first temperature and a reference pressure. The scaling law may be obtained by performing a two-parameter fit of the function, such as by identifying the slope and intercept of the scaling law. It is preferable to perform the calibration at the temperature and pressure approximately equal to the expected temperature and a second pressure. This method allows the determination of the mean chain length and the distribution of chain lengths of the constituents of the sample under investigation. From this information, the composition of the sample may be determined.

[0021] In a second embodiment, a method for determining the characteristics of a fluid sample is disclosed, wherein the calibration samples are subject to different temperatures and the reference pressure. In this case the scaling law becomes a function of mean chain length and temperature and it is no longer necessary to substantially match the temperature of the calibration samples to the expected temperature of the sample under investigation. Now the scaling law may be determined using a four parameter fit, as described in more detail below.

[0022] In a third embodiment, a method for determining the characteristics of a fluid sample is disclosed, comprising: (a) obtaining measurements (diffusion or relaxation measurements) of a plurality of calibration samples at a first temperature and a reference pressure, wherein the calibration samples have differing mean chain lengths; (b) determining the density of more than one pure alkane or mixtures of alkanes (not necessarily the same as the calibration samples) at the first temperature and the reference pressure, wherein density is determined as a function of mean chain length; (c) obtaining measurements (diffusion or relaxation measurements) of the sample under investigation at the first temperature and a second pressure; (d) determining the density of the sample under investigation at the first temperature and the second pressure; (e) applying the density function of (c) to the density measurements of (d) and using the measurements of (a) to determine the scaling law at the second pressure in terms of chain length; (f) applying the scaling law of (e) to the data of (c) to determine the composition of the sample under investigation. The density measurements of (b) may be obtained from standard look-up tables (such as the NIST webbook). This method can be used to determine the composition of the sample under investigation by determining the distribution of chain lengths of the constituents of the sample under investigation. It is noted that the density measurements of (b) can be any density, including, but not limited to, mass density, carbon density, and hydrogen density (more commonly known as the hydrogen index).

[0023] The fourth embodiment comprises a manipulation of the third embodiment, wherein a range of temperatures is accounted for. More specifically, a method for determining the characteristics of a fluid sample is disclosed, comprising: (a) obtaining measurements (diffusion or relaxation measurements) of a plurality of calibration samples at reference pressure and at more than one temperature, wherein the calibration samples have differing mean chain lengths; (b) determining the density of more than one pure alkane or mixtures of alkanes at the reference pressure and at a temperature within or near the range of temperatures in (a), wherein density is determined as a function of mean chain length; (c) obtaining measurements (diffusion or relaxation measurements) of the sample under investigation at a second pressure and at a temperature within or near the range of temperatures in (a); (c) determining the density of the sample under investigation at the second pressure and at a temperature within or near the range of temperatures in (a); (d) applying the density function of (c) to the density measurements of (d) and using the measurements of (a) to determine the scaling law at the second pressure in terms of chain length; (e) applying the scaling law of (e) to the data of (c) to determine the composition of the sample under investigation.

[0024] In a fifth embodiment, a method for determining the characteristics of a fluid sample is disclosed, comprising: (a) obtaining measurements (diffusion or relaxation measurements) of a plurality of calibration samples at a first temperature and a reference pressure, wherein the calibration samples have differing mean chain lengths; (b) obtaining measurements (diffusion or relaxation measurements) of a sample under investigation at the first temperature and at a second pressure; (c) determining the relationship of volume of one or more alkanes or mixtures of alkanes (not necessarily the calibration sample) to (i) the mean chain length at the first temperature and the reference pressure and (ii) the mean chain length at the first temperature and a second pressure; (d) determining the scaling law as a function of chain length, using the functions of (c) and the measurements of (a); (e) applying the scaling law of (d) with the measurements of (b) to determine the composition of the sample under investigation. The volumes of (c) may be obtained using standard look-up tables (such as the NIST webbook). Further, the volumes may be any volume, including, but not limited to, volume per mole (molar volume), volume per hydrogen atom, or volume per carbon atom.

[0025] The sixth embodiment is a manipulation of the fifth embodiment to account for various temperatures. More specifically, a method for determining the characteristics of a fluid sample, comprising: (a) obtaining measurements (diffusion or relaxation measurements) of a plurality of calibration samples at more than one temperature and a reference pressure, wherein the calibration samples have differing mean chain lengths; (b) obtaining measurements (diffusion or relaxation measurements) of a sample under investigation at a temperature within or near the range of temperatures of the measurements of (a) and at a second pressure; (c) determining the relationship of volume of alkanes or mixtures of alkanes to (i) the mean chain length at the reference pressure and (ii) the mean chain length at the second pressure; (d) determining the scaling law in terms chain length and temperature at the second pressure using the functions of (c) and the measurements of (a); (e) applying the scaling law of (d) with the measurements of (b) to determine the composition of the sample under investigation.

[0026] It is envisioned that these methods may be performed in a laboratory or at the point of sampling. For example, these methods may be particularly useful in the characterization of oilfields and may be used on samples obtained from the earth formation or within the earth formation.

[0027] Further features and applications of the present invention will become more readily apparent from the figures and detailed description that follows.

[0028]

[0029]

[0030]

[0031]

[0032] _{i}_{i}^{v }_{g}_{g }

[0033]

[0034]

[0035]

[0036] FIGS.

[0037]

[0038] _{1 }

[0039] ^{k}_{1}

[0040] _{i}^{k}_{1i }

[0041] _{i}^{v}_{i }

[0042] _{i}^{k}_{1i }_{eff}

[0043] _{T }

[0044] _{T }

[0045] FIGS. _{T }

[0046] _{i}^{v}_{i }

[0047] _{e}_{0}_{e}^{β}_{i}_{i }

[0048]

[0049] _{1}^{k }_{eff}

[0050] Modeling Oils

[0051] Properties that follow from Equation (2) suggest a method for fluid typing using diffusion data of oils. The ratio of diffusion coefficients within a given mixture gives the ratio of the “sizes” of the components. Furthermore, if it is known how the product D_{i}_{i }

[0052]

[0053] Polymer models may be used to describe alkanes. Doi et al.

[0054] where N is the chain length. For alkanes, N is equal to the number of carbon atoms. The exponent v is approximately equal to ½ if there are no excluded volume effects and ⅗ if there are excluded volume effects.

[0055] For alkanes found in oils, the chain lengths are usually too short to be perfectly described by the ideal gaussian chain, because the chains are stiffer than a true gaussian chain. For very long alkanes and polymethylene chains (with N≧100) where gaussian behavior is observed, the parameter l is about {square root}{square root over (6.7)} times the actual distance between carbon atoms. For shorter chains, l is not a constant. Instead, it varies with chain length, decreasing to the actual distance between carbon atoms for a chain length of two. However, even when other types of interactions between beads are used, many of the results for gaussian chains still apply, at least qualitatively.

[0056] Oils and liquid alkanes are melts. In melts, it is usually assumed that the hydrodynamic effects are screened out by the chains and that the excluded volume effects within a chain are balanced out by the excluded volume effects between differing chains. In that case, the melt is described by the Rouse model (shown in

[0057] where ξ is the friction constant for a bead and N is the number of beads in the chain. The Rouse model accounts for the gaussian interaction between nearest neighbors and each bead feeling the coefficient of friction, ξ, from the surrounding fluid.

[0058] For the shorter chains such as the alkanes, the hydrodynamic effects are not necessarily all screened out. The Zimm model adds the hydrodynamic effects to the Rouse model (see Zimm, J. Chem. Phys. 24:269 (1956) (incorporated by reference herein in its entirety)). In the Zimm model, the translational diffusion coefficient is given by

[0059] and, in the absence of excluded volume effects, the rotational diffusion coefficient is given by

[0060] where the radius of gyration is given by Equation (3). The viscosity η_{s }^{1/3}^{v}_{s }

[0061] Even though the chains encountered in most oils are short and therefore are not perfect gaussian chains, many ideas about the collective motion of the internal degrees of freedom inherent in polymer models still apply. As discussed below, many of the polymer results appear to apply to the alkanes, at least qualitatively, if not quantitatively. Accordingly, the chains follow similar scaling laws.

[0062] Based on the polymer models, it is expected that the diffusion coefficient of the i^{th }

[0063] where N_{i }_{0 }

[0064] According to the various polymer models, it is expected that 0.5≦v≦1. In the example presented herein, v≈0.7 (see the fit of the data in

[0065] The methane and ethane molecules (and, most likely, the alkanes up to and including pentane) are more appropriately described by the hard sphere model, so it is expected that their diffusion coefficients will have the form

[0066] where r_{i }_{0 }_{i }_{i}^{v }

_{m}

[0067] and for ethane

_{e}

[0068] Within a mixture, then, the ratio of the diffusion coefficients of any two components depends on the ratio of their radius or chain length to some power. In other words, if components 1 and 2 are oils, then their diffusion coefficients have the ratio

_{1}_{2}_{2}^{v}_{1}^{v }

[0069] regardless of any other properties of the mixture, such as its composition, temperature or pressure. Similarly, for a gas and an oil, the ratio would be

_{g}_{o}_{o}^{v}_{g }

[0070] These equations are similar to what one would expect reading Bearman, “Statistical Mechanical Theory of Diffusion Coefficients in Binary Liquid Solutions,” J. Chem. Phys. 32(5):1308-1313 (1959) (incorporated by reference herein in its entirety) for nearly ideal fluids where the molar volume of the fluids does not change much upon mixing. However, in the present case, the ratio depends on the effective radii of the molecules r_{i }^{v }

[0071] Expressing Equations (11) and (12) slightly differently, it is expected that D_{i}_{i}^{v }_{i}_{i }_{g}_{g }_{1}_{1}^{v }^{−5 }^{2}

[0072] For the alkanes, to a good approximation, the products D_{i}_{i}^{v }_{i}_{i }

[0073] where {overscore (N)} is the average chain length in the mixture, given by

[0074] and x_{i }^{th }

[0075] The free volume model for alkanes in von Meerwall et al. “Diffusion of Liquid n-Alkanes: Free-Volume and Density Effects,” J. Chem. Phys. 108(10):4299-4304 (1998) and von Meerwall et al. “Diffusion in Binary Liquid n-Alkane and Alkane-polyethylene Blends,” J. Chem. Phys. 111(2):750-757 (1999) (incorporated by reference herein in their entireties) can be used to give an explanation for this property. In addition to depending on the activation energy E_{a }

[0076] where ƒ(T,M) is the free volume fraction, or the volume that is unoccupied divided by the total volume, and M is the molecular weight of the chain. For this application, chain length N and molecular weight M can be used interchangeably (see Equation (29) below) The model in von Meerwall et al. (1998), see

[0077] The function g({overscore (N)}) then depends on the constituents only through the average chain length. Note that the molecular average is used, given by Equation (14) while von Meerwall et al. (1999) instead stipulate that it depends on an average M* defined by

[0078] where v_{i }

[0079] The data for alkanes in Douglas et al. “Diffusion in Paraffin Hydrocarbons,” J. Phys. Chem. 62:1102-1107 (1958) (incorporated by reference herein in its entirety) and for mixtures of alkanes in Van Geet et al., Lo et al. (1998), and Freedman et al. appear to fit Equation (13) quite well as shown in

[0080] In addition, for the live oils in Helbaek et al., shown in _{0}_{0}^{v }_{g}_{g }

[0081] The free volume theory gives a complicated functional dependence of D_{i}_{i}^{v }

[0082] where A and β are both slowly varying functions of temperature and pressure. The diffusion coefficient for the i^{th }

[0083] For example, the data for alkanes and mixtures of alkanes (and also mixtures with squalene) at room temperature and atmospheric pressure is plotted in ^{−5 }^{2}

[0084] For live oils, D_{i}_{i }

[0085] Changing N_{i }_{i}

[0086] The relationship between chain length and diffusion coefficients can be used for fluid typing. At a particular pressure and temperature, A and β can be determined by several measurements on known fluids or liquids. Because the function g({overscore (N)}) depends only on the mean chain length, it does not matter which particular alkanes or mixtures are used. In this way, g({overscore (N)}) may be obtained for the values of temperature and pressure of interest. For example, in

[0087] Using Equations (18) or (19) and (14), the mean chain length can be determined from the measured distribution function ƒ(D_{i}

[0088] Once the mean chain length is determined, the distribution of chain lengths may be determined using

[0089] For mixtures of alkanes, the regularized inverse Laplace transform of the NMR data will give a distribution of diffusion coefficients, which can be inverted for the chain length distribution, as described above. One caveat to using the inverse Laplace transform is that it will still give a relatively broad distribution, even in the case where there is only one or two diffusion coefficients, as shown in

[0090] FIGS.

[0091] FIGS.

[0092] The polymer model can also be used to find the viscosity of a mixture, given the distribution of diffusion coefficients. For a polymer, according to the Rouse and Zimm models, the viscosity is related to the rotational diffusion coefficient D_{R }

[0093] In this equation, c is the number of segments per unit volume and is related to the density ρ by c=ηN/M, where M is the mass of the chain. The constant b′ depends on whether the Rouse or Zimm model is used. For both the Rouse and Zimm models without excluded volume effects, the rotational and translational diffusion coefficients are related by

[0094] Again, the constant of proportionality depends on whether the Rouse or Zimm model is used. Combining equations (21) and (22) gives the relation between the viscosity and the translational diffusion coefficient:

^{2}

[0095] where b is a constant that depends on which model is used. For the Rouse model it is {fraction (1/36)} and for the Zimm model it is 0.0833.

[0096] Note that the product ηD/T is independent or nearly independent of chain length (see Lo et al. (2000) and Freedman et al). This would not be the case for hard spheres, where the product would be expected to scale with the chain length. Instead, in the polymer models the chain length scaling drops out due to the “anamolous” dependence on chain length of both diffusion coefficients.

[0097] These equations can be checked more quantitatively by comparing the predictions for the values of ηD/T from the polymer models with those found experimentally. For alkanes and refined oils, ηD/T was found to be 3.90^{−9 }^{2}^{−8 }^{2}^{3 }^{−8 }^{2}^{3}^{−8 }^{2}^{−8 }

[0098] In a mixture, according to the polymer models (see Ferry), the viscosity is just a sum of the viscosity of each component in the mixture, weighted by the number of molecules of that component per unit volume. Thus, the total viscosity is

[0099] The relation between the translational and rotational diffusion coefficients then gives

[0100] where y_{i }^{th }

[0101] In

[0102] Accounting for the Effects of Pressure

[0103] It may be preferred that the above model be refined to account for the effects of pressure and temperature on NMR measurements, such as diffusion coefficient and relaxation time. As will be shown below, diffusion coefficient D_{i }_{1,2i }_{i }_{1,2i }

[0104] As discussed above, the free volume fraction depends on the volume/end v_{e}_{sf}_{so}

[0105] The free volume fraction can be written in terms of the mean chain length and various volumes as follows:

[0106] Density may also be written in terms of these parameters. For a pure fluid, the density is given by:

[0107] where v_{T }

[0108] the number of grams/mole is given by:

[0109] and the expression for the density becomes

[0110] Next, in both the expression for the free volume fraction and the density, the pressure and {overscore (N)}-dependent parts may be separated from the rest as follows:

[0111] Thus, in both the density and the free volume fraction, the only dependence on the pressure and mean chain length is through the combination

[0112] In other words, the density and free volume fraction may be written as:

[0113] Then the free volume fraction can be written in terms of the density as follows:

[0114] With the assumption that v_{so }_{i}_{i}^{v }_{1i}_{1i}^{k }_{2i}_{2i}^{k }

[0115] In _{1 }

[0116] ^{k}_{1 }

[0117] _{1i}_{i}^{−k}_{1i}_{i}^{−k }

[0118] Next, the dependence of the diffusion coefficient on density is analyzed. In _{i}_{i}^{v }

[0119] The data collapse reasonably well to a single line. In fact, over the entire range, the agreement between hexane and octane is quite remarkable. However, as the density nears 0.75 g/cm^{3}

[0120] Accordingly, the free volume and, hence, the diffusion coefficients and relaxation times are functions of the density. Accordingly, if the scaling laws are known at the temperature of interest and at one reference pressure, then the relationship between diffusion coefficients and composition can be determined at any pressure, as long as the density of the oil is known. This means that chain length distribution can be determined from a measurement of both density and the diffusion or relaxation distribution.

[0121] For the following discussion, it is assumed that the equation for the diffusion coefficient at atmospheric pressure P_{0 }_{eff }

[0122] Using Equation (30) for density,

[0123] where M=14.06N_{eff}_{s }_{e }

[0124] Equations of von Meerwall et al. (1998) may be used to determine the values of v_{s }_{e}

_{0}_{∞}_{E}^{−1 }

[0125] where

_{∞}^{3}

[0126] and

_{E}^{3}

[0127] Setting Equation (39) equal to Equation (37) and substituting in the value for the mass M, v_{s }_{e }

_{s}^{−1}_{∞}

[0128] and the extra volume/end is given by

_{e}^{−1}_{∞}_{E }

[0129] Next, the diffusion coefficient is calculated at elevated pressure. At temperature T, the scaled diffusion coefficient is a function of density, so that

_{i}^{v}_{i}_{i}^{v}_{i}_{eff}_{0}

[0130] Over some range of temperatures and chain lengths, the diffusion coefficient at atmospheric pressure has the form

_{i}_{eff}_{0}_{0}_{i}^{−v}_{eff}^{−β(T,P}_{0}^{) }

[0131] Above it was shown (see discussion of Equation (18)) that A(T,P_{0}^{−5 }^{2}_{0}_{0}_{0}

_{i}_{0}_{i}^{−v}_{eff}^{−β(T,P}_{0}^{) }

[0132] where N_{eff }

[0133] A similar calculation for the relaxation times yields

_{1i}_{2i}_{0}_{i}^{−k}_{eff}^{−γ(T,P}_{0}^{) }

[0134] where B(T,P_{0}_{0}_{1 }_{eff}_{1 }

[0135] For live oils, the reference pressure is preferably not equivalent to the atmospheric pressure for the following reasons: (1) the scaling law is extrapolated beyond the range where it was fit; (2) the scaling law is applied to a regime of short chains which is not truly physical because at atmospheric pressure the alkanes are no longer liquids for chain lengths less than C_{eff}_{0}

[0136] Instead, the scaling law for live oils should be determined using an effective chain length that is defined at an elevated pressure P, where the full range of N_{eff}_{eff}_{eff}_{0}_{0 }_{eff}_{0}_{0 }_{s }_{e}_{eff}_{0}_{0 }

[0137] This method works when N_{eff }_{1,2 }_{eff }

[0138] First, the pressure dependence of the molar volume will be examined. The molar volume is given by

[0139] where v_{e }_{s }_{so }_{sf}_{e }_{s }_{e }_{s }_{T}

[0140] In

[0141] The values for the slope v_{s }_{e }

Temperature | Pressure | Slope ν_{s} | Intercept 2ν_{e} | |

(° C.) | (MPa) | (cm^{3} | (cm^{3} | |

25° | 0.1 | 16.27 | 33.20 | |

0.14 | 16.32 | 32.61 | ||

20.7 | 16.18 | 30.00 | ||

41.4 | 16.01 | 28.45 | ||

50.0 | 16.07 | 27.06 | ||

85° | 0.1 | 16.91 | 40.49 | |

0.14 | 16.84 | 40.84 | ||

20.7 | 16.74 | 35.56 | ||

41.4 | 16.54 | 33.01 | ||

[0142] _{s }_{e }

Temperature | Pressure | Slope ν_{s} | Intercept 2ν_{e} | |

(° C.) | (MPa) | (cm^{3} | (cm^{3} | |

30° | 0.1 | 16.33 | 33.81 | |

30.0 | 16.08 | 30.31 | ||

40.0 | 16.09 | 28.79 | ||

50.0 | 16.09 | 27.53 | ||

50° | 0.1 | 16.54 | 36.24 | |

23.4 | 16.12 | 34.08 | ||

50.9 | 16.20 | 29.19 | ||

60° C. | 0.1 | 16.65 | 37.45 | |

30.0 | 16.22 | 33.72 | ||

40.0 | 16.25 | 31.74 | ||

50.0 | 16.27 | 30.12 | ||

[0143] The densities for live mixtures of C_{s }_{e }

[0144] In both Table 1 and 2, the slope v_{s }_{e }_{s }_{e }_{s }_{so }

[0145] In order to determine the pressure dependence on the diffusion coefficients, the effective chain length N_{eff }_{eff }_{0 }_{0 }_{eff }_{eff}

[0146] The expression for ρ in Equation (32) is now used to solve for N_{eff}

[0147] Based on the values of v_{s }_{e }_{s }_{e}_{s}_{s}_{0}_{e}

[0148] Now the diffusion coefficient at pressure P may be determined. As before,

[0149] From the scaling law for D(N_{eff}_{0}

[0150] Substituting in the expression for N_{eff}

[0151] In this way, the scaling law as a function of {overscore (N)} for D_{i }

[0152] The pressure dependence for the relaxation times can be found in a similar way, and has the form

[0153] If information about the density of alkanes at the desired temperature T and at both the desired pressure P and the reference pressure P_{0 }_{e}_{0}_{e}_{0}

[0154] An example is provided in _{e}_{0}_{e}_{0}_{0}

[0155] Thus, once the scaling law at atmospheric pressure is known as well as some densities at atmospheric pressure and at 50 MPa, a reasonably good fit to the diffusion coefficients at 50 MPa may be obtained with no fitting parameters.

[0156] One of the consequences of the pressure-independence of β(T) is that, on a log plot, the curves for diffusion coefficients as a function of temperature or plots of distributions of diffusion coefficients will all lie parallel to each other as the pressure is changed. However, once the pressure becomes very high, the parameter, β(T) will have some pressure dependence, as found by Vardag et al., for pressure above about 100 MPa (above about 100 MPa, the slope starts to change). This presumably is an indication that the occupied volume per segment is also affected by pressure at these high pressures. As described below, it is probably also an indication that the diffusion coefficient depends on pressure as well as density at these higher pressures. (This, in turn, should indicate that at high pressures the occupied volume depends on pressure.)

[0157] The fact that the exponent, β(T) is independent of pressure (at pressures that are not extremely high) follows from the fact that the free volume from the ends is much larger than the change in volume of the segments as the pressure is changed. It also turns out that it is a consequence of requiring the diffusion coefficient D both to depend on pressure and chain length only through density and to follow scaling laws at any pressure. For example, if Equation (51) is substituted into Equation (54) for the effective chain length N_{eff }_{s }_{e}

[0158] If D({overscore (N)}) is required to obey a strict scaling law in {overscore (N)}, the relation in Equation (58) is only possible if the expression in the denominator is replaced with 2v_{e}_{s}_{s}_{0}_{e}_{i }

[0159] For short chains, the molar volume is no longer a linear function of chain length. At the temperatures of _{0 }

_{i}_{0}_{i}_{eff(P}_{0}_{)}^{−β(T,P}_{0}^{) }

[0160] where r_{i}_{i}^{84 }_{i }_{eff }_{s}_{s}_{0}_{e}

[0161] In all the examples above, the extra free energy from the edges v_{e }_{e}_{e}_{0}

[0162] The fit to a scaling law does not seem to be that sensitive to this approximation, as will be described below.

[0163] _{e}_{0}_{s}^{β}_{0}

[0164] is plotted versus the actual mean chain length of the alkane or mixture. The reference pressure P_{0 }_{e}_{0}_{e}

[0165] Note that the quantity on the right hand side of Equation (62) depends only on the reference pressure. Thus, the data points at all four pressures should collapse to a single line. As can be seen in

[0166] Accordingly, if the scaling law is known at some temperature and reference pressure and if the dependence on chain length of the molar volume is known at the reference pressure and any pressure P, then the diffusion coefficients and the relaxation items may be calculated at pressure P. Having determined the pressure dependence, now the temperature dependence is considered.

[0167] Accounting for the Effects of Temperature

[0168] The power law dependence on chain length is combined with the Ahrrenius temperature dependence to determine how diffusion and relaxation depend on temperature and chain length. As above, the discussion is focused on diffusion coefficients, but may be similarly applied to relaxation time.

[0169] According to Equation (18), the diffusion coefficient follows a scaling law of the form:

[0170] where A(T,P) and β(T) depend on temperature. Alternatively, for pure substances, the diffusion coefficient D has been found to have an Arrhenius temperature dependence of the form

^{−E}^{a}^{(N)}

[0171] where the activation energy E_{a}

_{a}

[0172] for some temperature-independent coefficients b and d. This was in fact found in Ertl et al. and von Meerwall et al. (1998). The diffusion coefficient can then be written in terms of four temperature-independent parameters a, b, c, and d in the form

[0173] In other words, the exponent β(T) in the scaling law is given by

[0174] and the coefficient A(T,P) is given by

[0175] Because A depends on pressure, the parameters a and b can also depend on pressure. However, since β is independent of pressure, c and d should also be independent of pressure. The temperature dependence for the relaxation times can be found in a similar manner, with the result

[0176] where a′(P), b′(P), c′, and d′ are temperature-independent parameters.

[0177] To illustrate this, data for diffusion coefficients for pure alkanes taken at a wide range of temperatures and at atmospheric pressure (or saturation vapor pressure) are shown. The diffusion coefficients as a function of reciprocal temperature are plotted in

[0178] The values of the four fitted parameters were

[0179] a=−6.3326

[0180] b=143.6869

[0181] c=−0.2442

[0182] d=588.4961

[0183] when the coefficient A(T,P) is given in 10^{−5 }^{2}^{−5 }^{2}^{−5 }^{2}^{−5 }^{2}

[0184] In order to look at live oils, the data is also fit to Equation (66), with the result

[0185] a=−5.7256

[0186] b=−212.9887

[0187] c=−0.4636

[0188] d=705.1817

[0189] With these parameters, the fit to the data looks almost identical to the fit with the parameters for the first scaling law. However, as determined above, even though this equation fits the data for the dead oils quite well, it still does not extrapolate well to smaller chain lengths for the live oils.

[0190] Next, the T_{1 }_{eff }

[0191] a′=−5.75

[0192] b′=−227

[0193] c′=−1.43

[0194] d′=755

[0195] when T_{1 }

[0196] It is noted that while the above applications relate to oil applications, the method may be adapted for other applications including the medical and food preparation industries, for example.

[0197] While the invention has been described herein with reference to certain examples and embodiments, it will be evident that various modifications and changes may be made to the embodiments described above without departing from the scope and spirit of the invention as set forth in the claims.